The common area of the curves `y = sqrtx` and `x = sqrty` is equal to

Options

A. (a) 3 sq. units
B. (b) `(5)/(3)` sq. units
C. (c) `(1)/(3)` sq. units
D. (d) `(2)/(3)` sq. units

Answer

To find the common area of the curves \( y = \sqrt{x} \) and \( x = \sqrt{y} \), we will follow these steps:

### Step 1: Identify the curves and their intersection points
The given curves are:
1. \( y = \sqrt{x} \)
2. \( x = \sqrt{y} \)

To find the points of intersection, we can set \( y = \sqrt{x} \) equal to \( x = \sqrt{y} \).

Substituting \( y = \sqrt{x} \) into \( x = \sqrt{y} \):
\[ x = \sqrt{\sqrt{x}} = x^{1/4} \]
Now, squaring both sides gives:
\[ x^2 = x^{1/2} \]
Rearranging this, we have:
\[ x^2 - x^{1/2} = 0 \]
Factoring out \( x^{1/2} \):
\[ x^{1/2}(x^{3/2} - 1) = 0 \]
This gives us two solutions:
1. \( x^{1/2} = 0 \) → \( x = 0 \)
2. \( x^{3/2} - 1 = 0 \) → \( x = 1 \)

Thus, the points of intersection are \( (0, 0) \) and \( (1, 1) \).

### Step 2: Set up the integral for the area
The area between the curves from \( x = 0 \) to \( x = 1 \) can be calculated using the formula:
\[ \text{Area} = \int_{a}^{b} (y_2 - y_1) \, dx \]
Here, \( y_2 = \sqrt{x} \) and \( y_1 = x^2 \) (since \( x = \sqrt{y} \) implies \( y = x^2 \)).

### Step 3: Calculate the area
The area can be expressed as:
\[ \text{Area} = \int_{0}^{1} (\sqrt{x} - x^2) \, dx \]

Now, we will evaluate the integral:
\[ \text{Area} = \int_{0}^{1} x^{1/2} \, dx - \int_{0}^{1} x^2 \, dx \]

Calculating these integrals separately:
1. For \( \int x^{1/2} \, dx \):
\[ \int x^{1/2} \, dx = \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2} \]
Evaluating from 0 to 1:
\[ \left[ \frac{2}{3} x^{3/2} \right]_{0}^{1} = \frac{2}{3}(1) - \frac{2}{3}(0) = \frac{2}{3} \]

2. For \( \int x^2 \, dx \):
\[ \int x^2 \, dx = \frac{x^3}{3} \]
Evaluating from 0 to 1:
\[ \left[ \frac{x^3}{3} \right]_{0}^{1} = \frac{1}{3} - 0 = \frac{1}{3} \]

### Step 4: Combine the results
Now, substituting back into the area formula:
\[ \text{Area} = \frac{2}{3} - \frac{1}{3} = \frac{1}{3} \]

### Final Answer
Thus, the common area of the curves \( y = \sqrt{x} \) and \( x = \sqrt{y} \) is:
\[ \text{Area} = \frac{1}{3} \text{ square units} \]